! ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
! 
! Numerical Integrator (Time-Stepping) File
! 
! Generated by KPP-2.2.3 symbolic chemistry Kinetics PreProcessor
!       (http://www.cs.vt.edu/~asandu/Software/KPP)
! KPP is distributed under GPL, the general public licence
!       (http://www.gnu.org/copyleft/gpl.html)
! (C) 1995-1997, V. Damian & A. Sandu, CGRER, Univ. Iowa
! (C) 1997-2005, A. Sandu, Michigan Tech, Virginia Tech
!     With important contributions from:
!        M. Damian, Villanova University, USA
!        R. Sander, Max-Planck Institute for Chemistry, Mainz, Germany
! 
! File                 : aromatics_kpp_Integrator.f90
! Time                 : Fri Jun 12 02:54:05 2020
! Working directory    : /n/home08/kbates/Aromatics/MCM_update_chamber
! Equation file        : aromatics_kpp.kpp
! Output root filename : aromatics_kpp
! 
! ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~



! ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
! 
! INTEGRATE - Integrator routine
!   Arguments :
!      TIN       - Start Time for Integration
!      TOUT      - End Time for Integration
! 
! ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

!~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~!
!  Rosenbrock - Implementation of several Rosenbrock methods:             !
!               * Ros2                                                    !
!               * Ros3                                                    !
!               * Ros4                                                    !
!               * Rodas3                                                  !
!               * Rodas4                                                  !
!  By default the code employs the KPP sparse linear algebra routines     !
!  Compile with -DFULL_ALGEBRA to use full linear algebra (LAPACK)        !
!                                                                         !
!    (C)  Adrian Sandu, August 2004                                       !
!    Virginia Polytechnic Institute and State University                  !
!    Contact: sandu@cs.vt.edu                                             !
!    Revised by Philipp Miehe and Adrian Sandu, May 2006                  !                               !
!    This implementation is part of KPP - the Kinetic PreProcessor        !
!~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~!

MODULE aromatics_kpp_Integrator

  USE aromatics_kpp_Parameters, ONLY: NVAR, NFIX, NSPEC, LU_NONZERO
  USE aromatics_kpp_Global
  IMPLICIT NONE
  PUBLIC
  SAVE
  
!~~~>  Statistics on the work performed by the Rosenbrock method
  INTEGER, PARAMETER :: Nfun=1, Njac=2, Nstp=3, Nacc=4, &
                        Nrej=5, Ndec=6, Nsol=7, Nsng=8, &
                        Ntexit=1, Nhexit=2, Nhnew = 3

CONTAINS

SUBROUTINE INTEGRATE( TIN, TOUT, &
  ICNTRL_U, RCNTRL_U, ISTATUS_U, RSTATUS_U, IERR_U )

   IMPLICIT NONE

   REAL(kind=dp), INTENT(IN) :: TIN  ! Start Time
   REAL(kind=dp), INTENT(IN) :: TOUT ! End Time
   ! Optional input parameters and statistics
   INTEGER,       INTENT(IN),  OPTIONAL :: ICNTRL_U(20)
   REAL(kind=dp), INTENT(IN),  OPTIONAL :: RCNTRL_U(20)
   INTEGER,       INTENT(OUT), OPTIONAL :: ISTATUS_U(20)
   REAL(kind=dp), INTENT(OUT), OPTIONAL :: RSTATUS_U(20)
   INTEGER,       INTENT(OUT), OPTIONAL :: IERR_U

   REAL(kind=dp) :: RCNTRL(20), RSTATUS(20)
   INTEGER       :: ICNTRL(20), ISTATUS(20), IERR

   INTEGER, SAVE :: Ntotal = 0

   ICNTRL(:)  = 0
   RCNTRL(:)  = 0.0_dp
   ISTATUS(:) = 0
   RSTATUS(:) = 0.0_dp

    !~~~> fine-tune the integrator:
   ICNTRL(1) = 0	! 0 - non-autonomous, 1 - autonomous
   ICNTRL(2) = 0	! 0 - vector tolerances, 1 - scalars

   ! If optional parameters are given, and if they are >0, 
   ! then they overwrite default settings. 
   IF (PRESENT(ICNTRL_U)) THEN
     WHERE(ICNTRL_U(:) > 0) ICNTRL(:) = ICNTRL_U(:)
   END IF
   IF (PRESENT(RCNTRL_U)) THEN
     WHERE(RCNTRL_U(:) > 0) RCNTRL(:) = RCNTRL_U(:)
   END IF


   CALL Rosenbrock(NVAR,VAR,TIN,TOUT,   &
         ATOL,RTOL,                &
         RCNTRL,ICNTRL,RSTATUS,ISTATUS,IERR)

   !~~~> Debug option: show no of steps
   ! Ntotal = Ntotal + ISTATUS(Nstp)
   ! PRINT*,'NSTEPS=',ISTATUS(Nstp),' (',Ntotal,')','  O3=', VAR(ind_O3)

   STEPMIN = RSTATUS(Nhexit)
   ! if optional parameters are given for output they 
   ! are updated with the return information
   IF (PRESENT(ISTATUS_U)) ISTATUS_U(:) = ISTATUS(:)
   IF (PRESENT(RSTATUS_U)) RSTATUS_U(:) = RSTATUS(:)
   IF (PRESENT(IERR_U))    IERR_U       = IERR

END SUBROUTINE INTEGRATE

!~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
SUBROUTINE Rosenbrock(N,Y,Tstart,Tend, &
           AbsTol,RelTol,              &
           RCNTRL,ICNTRL,RSTATUS,ISTATUS,IERR)
!~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
!
!    Solves the system y'=F(t,y) using a Rosenbrock method defined by:
!
!     G = 1/(H*gamma(1)) - Jac(t0,Y0)
!     T_i = t0 + Alpha(i)*H
!     Y_i = Y0 + \sum_{j=1}^{i-1} A(i,j)*K_j
!     G * K_i = Fun( T_i, Y_i ) + \sum_{j=1}^S C(i,j)/H * K_j +
!         gamma(i)*dF/dT(t0, Y0)
!     Y1 = Y0 + \sum_{j=1}^S M(j)*K_j
!
!    For details on Rosenbrock methods and their implementation consult:
!      E. Hairer and G. Wanner
!      "Solving ODEs II. Stiff and differential-algebraic problems".
!      Springer series in computational mathematics, Springer-Verlag, 1996.
!    The codes contained in the book inspired this implementation.
!
!    (C)  Adrian Sandu, August 2004
!    Virginia Polytechnic Institute and State University
!    Contact: sandu@cs.vt.edu
!    Revised by Philipp Miehe and Adrian Sandu, May 2006                  
!    This implementation is part of KPP - the Kinetic PreProcessor
!~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
!
!~~~>   INPUT ARGUMENTS:
!
!-     Y(N)    = vector of initial conditions (at T=Tstart)
!-    [Tstart,Tend]  = time range of integration
!     (if Tstart>Tend the integration is performed backwards in time)
!-    RelTol, AbsTol = user precribed accuracy
!- SUBROUTINE  Fun( T, Y, Ydot ) = ODE function,
!                       returns Ydot = Y' = F(T,Y)
!- SUBROUTINE  Jac( T, Y, Jcb ) = Jacobian of the ODE function,
!                       returns Jcb = dFun/dY
!-    ICNTRL(1:20)    = integer inputs parameters
!-    RCNTRL(1:20)    = real inputs parameters
!~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
!
!~~~>     OUTPUT ARGUMENTS:
!
!-    Y(N)    -> vector of final states (at T->Tend)
!-    ISTATUS(1:20)   -> integer output parameters
!-    RSTATUS(1:20)   -> real output parameters
!-    IERR            -> job status upon return
!                        success (positive value) or
!                        failure (negative value)
!~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
!
!~~~>     INPUT PARAMETERS:
!
!    Note: For input parameters equal to zero the default values of the
!       corresponding variables are used.
!
!    ICNTRL(1) = 1: F = F(y)   Independent of T (AUTONOMOUS)
!              = 0: F = F(t,y) Depends on T (NON-AUTONOMOUS)
!
!    ICNTRL(2) = 0: AbsTol, RelTol are N-dimensional vectors
!              = 1: AbsTol, RelTol are scalars
!
!    ICNTRL(3)  -> selection of a particular Rosenbrock method
!        = 0 :    Rodas3 (default)
!        = 1 :    Ros2
!        = 2 :    Ros3
!        = 3 :    Ros4
!        = 4 :    Rodas3
!        = 5 :    Rodas4
!
!    ICNTRL(4)  -> maximum number of integration steps
!        For ICNTRL(4)=0) the default value of 100000 is used
!
!    RCNTRL(1)  -> Hmin, lower bound for the integration step size
!          It is strongly recommended to keep Hmin = ZERO
!    RCNTRL(2)  -> Hmax, upper bound for the integration step size
!    RCNTRL(3)  -> Hstart, starting value for the integration step size
!
!    RCNTRL(4)  -> FacMin, lower bound on step decrease factor (default=0.2)
!    RCNTRL(5)  -> FacMax, upper bound on step increase factor (default=6)
!    RCNTRL(6)  -> FacRej, step decrease factor after multiple rejections
!                          (default=0.1)
!    RCNTRL(7)  -> FacSafe, by which the new step is slightly smaller
!         than the predicted value  (default=0.9)
!~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
!
!
!    OUTPUT ARGUMENTS:
!    -----------------
!
!    T           -> T value for which the solution has been computed
!                     (after successful return T=Tend).
!
!    Y(N)        -> Numerical solution at T
!
!    IDID        -> Reports on successfulness upon return:
!                    = 1 for success
!                    < 0 for error (value equals error code)
!
!    ISTATUS(1)  -> No. of function calls
!    ISTATUS(2)  -> No. of jacobian calls
!    ISTATUS(3)  -> No. of steps
!    ISTATUS(4)  -> No. of accepted steps
!    ISTATUS(5)  -> No. of rejected steps (except at very beginning)
!    ISTATUS(6)  -> No. of LU decompositions
!    ISTATUS(7)  -> No. of forward/backward substitutions
!    ISTATUS(8)  -> No. of singular matrix decompositions
!
!    RSTATUS(1)  -> Texit, the time corresponding to the
!                     computed Y upon return
!    RSTATUS(2)  -> Hexit, last accepted step before exit
!    RSTATUS(3)  -> Hnew, last predicted step (not yet taken)
!                   For multiple restarts, use Hnew as Hstart 
!                     in the subsequent run
!
!~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

  USE aromatics_kpp_Parameters
  USE aromatics_kpp_LinearAlgebra
  IMPLICIT NONE

!~~~>  Arguments
   INTEGER,       INTENT(IN)    :: N
   REAL(kind=dp), INTENT(INOUT) :: Y(N)
   REAL(kind=dp), INTENT(IN)    :: Tstart,Tend
   REAL(kind=dp), INTENT(IN)    :: AbsTol(N),RelTol(N)
   INTEGER,       INTENT(IN)    :: ICNTRL(20)
   REAL(kind=dp), INTENT(IN)    :: RCNTRL(20)
   INTEGER,       INTENT(INOUT) :: ISTATUS(20)
   REAL(kind=dp), INTENT(INOUT) :: RSTATUS(20)
   INTEGER, INTENT(OUT)   :: IERR
!~~~>  Parameters of the Rosenbrock method, up to 6 stages
   INTEGER ::  ros_S, rosMethod
   INTEGER, PARAMETER :: RS2=1, RS3=2, RS4=3, RD3=4, RD4=5, RG3=6
   REAL(kind=dp) :: ros_A(15), ros_C(15), ros_M(6), ros_E(6), &
                    ros_Alpha(6), ros_Gamma(6), ros_ELO
   LOGICAL :: ros_NewF(6)
   CHARACTER(LEN=12) :: ros_Name
!~~~>  Local variables
   REAL(kind=dp) :: Roundoff, FacMin, FacMax, FacRej, FacSafe
   REAL(kind=dp) :: Hmin, Hmax, Hstart
   REAL(kind=dp) :: Texit
   INTEGER       :: i, UplimTol, Max_no_steps
   LOGICAL       :: Autonomous, VectorTol
!~~~>   Parameters
   REAL(kind=dp), PARAMETER :: ZERO = 0.0_dp, ONE  = 1.0_dp
   REAL(kind=dp), PARAMETER :: DeltaMin = 1.0E-5_dp

!~~~>  Initialize statistics
   ISTATUS(1:8) = 0
   RSTATUS(1:3) = ZERO

!~~~>  Autonomous or time dependent ODE. Default is time dependent.
   Autonomous = .NOT.(ICNTRL(1) == 0)

!~~~>  For Scalar tolerances (ICNTRL(2).NE.0)  the code uses AbsTol(1) and RelTol(1)
!   For Vector tolerances (ICNTRL(2) == 0) the code uses AbsTol(1:N) and RelTol(1:N)
   IF (ICNTRL(2) == 0) THEN
      VectorTol = .TRUE.
      UplimTol  = N
   ELSE
      VectorTol = .FALSE.
      UplimTol  = 1
   END IF

!~~~>   Initialize the particular Rosenbrock method selected
   SELECT CASE (ICNTRL(3))
     CASE (1)
       CALL Ros2
     CASE (2)
       CALL Ros3
     CASE (3)
       CALL Ros4
     CASE (0,4)
       CALL Rodas3
     CASE (5)
       CALL Rodas4
     CASE (6)
       CALL Rang3
     CASE DEFAULT
       PRINT * , 'Unknown Rosenbrock method: ICNTRL(3)=',ICNTRL(3) 
       CALL ros_ErrorMsg(-2,Tstart,ZERO,IERR)
       RETURN
   END SELECT

!~~~>   The maximum number of steps admitted
   IF (ICNTRL(4) == 0) THEN
      Max_no_steps = 200000
   ELSEIF (ICNTRL(4) > 0) THEN
      Max_no_steps=ICNTRL(4)
   ELSE
      PRINT * ,'User-selected max no. of steps: ICNTRL(4)=',ICNTRL(4)
      CALL ros_ErrorMsg(-1,Tstart,ZERO,IERR)
      RETURN
   END IF

!~~~>  Unit roundoff (1+Roundoff>1)
   Roundoff = WLAMCH('E')

!~~~>  Lower bound on the step size: (positive value)
   IF (RCNTRL(1) == ZERO) THEN
      Hmin = ZERO
   ELSEIF (RCNTRL(1) > ZERO) THEN
      Hmin = RCNTRL(1)
   ELSE
      PRINT * , 'User-selected Hmin: RCNTRL(1)=', RCNTRL(1)
      CALL ros_ErrorMsg(-3,Tstart,ZERO,IERR)
      RETURN
   END IF
!~~~>  Upper bound on the step size: (positive value)
   IF (RCNTRL(2) == ZERO) THEN
      Hmax = ABS(Tend-Tstart)
   ELSEIF (RCNTRL(2) > ZERO) THEN
      Hmax = MIN(ABS(RCNTRL(2)),ABS(Tend-Tstart))
   ELSE
      PRINT * , 'User-selected Hmax: RCNTRL(2)=', RCNTRL(2)
      CALL ros_ErrorMsg(-3,Tstart,ZERO,IERR)
      RETURN
   END IF
!~~~>  Starting step size: (positive value)
   IF (RCNTRL(3) == ZERO) THEN
      Hstart = MAX(Hmin,DeltaMin)
   ELSEIF (RCNTRL(3) > ZERO) THEN
      Hstart = MIN(ABS(RCNTRL(3)),ABS(Tend-Tstart))
   ELSE
      PRINT * , 'User-selected Hstart: RCNTRL(3)=', RCNTRL(3)
      CALL ros_ErrorMsg(-3,Tstart,ZERO,IERR)
      RETURN
   END IF
!~~~>  Step size can be changed s.t.  FacMin < Hnew/Hold < FacMax
   IF (RCNTRL(4) == ZERO) THEN
      FacMin = 0.2_dp
   ELSEIF (RCNTRL(4) > ZERO) THEN
      FacMin = RCNTRL(4)
   ELSE
      PRINT * , 'User-selected FacMin: RCNTRL(4)=', RCNTRL(4)
      CALL ros_ErrorMsg(-4,Tstart,ZERO,IERR)
      RETURN
   END IF
   IF (RCNTRL(5) == ZERO) THEN
      FacMax = 6.0_dp
   ELSEIF (RCNTRL(5) > ZERO) THEN
      FacMax = RCNTRL(5)
   ELSE
      PRINT * , 'User-selected FacMax: RCNTRL(5)=', RCNTRL(5)
      CALL ros_ErrorMsg(-4,Tstart,ZERO,IERR)
      RETURN
   END IF
!~~~>   FacRej: Factor to decrease step after 2 succesive rejections
   IF (RCNTRL(6) == ZERO) THEN
      FacRej = 0.1_dp
   ELSEIF (RCNTRL(6) > ZERO) THEN
      FacRej = RCNTRL(6)
   ELSE
      PRINT * , 'User-selected FacRej: RCNTRL(6)=', RCNTRL(6)
      CALL ros_ErrorMsg(-4,Tstart,ZERO,IERR)
      RETURN
   END IF
!~~~>   FacSafe: Safety Factor in the computation of new step size
   IF (RCNTRL(7) == ZERO) THEN
      FacSafe = 0.9_dp
   ELSEIF (RCNTRL(7) > ZERO) THEN
      FacSafe = RCNTRL(7)
   ELSE
      PRINT * , 'User-selected FacSafe: RCNTRL(7)=', RCNTRL(7)
      CALL ros_ErrorMsg(-4,Tstart,ZERO,IERR)
      RETURN
   END IF
!~~~>  Check if tolerances are reasonable
    DO i=1,UplimTol
      IF ( (AbsTol(i) <= ZERO) .OR. (RelTol(i) <= 10.0_dp*Roundoff) &
         .OR. (RelTol(i) >= 1.0_dp) ) THEN
        PRINT * , ' AbsTol(',i,') = ',AbsTol(i)
        PRINT * , ' RelTol(',i,') = ',RelTol(i)
        CALL ros_ErrorMsg(-5,Tstart,ZERO,IERR)
        RETURN
      END IF
    END DO


!~~~>  CALL Rosenbrock method
   CALL ros_Integrator(Y, Tstart, Tend, Texit,   &
        AbsTol, RelTol,                          &
!  Integration parameters
        Autonomous, VectorTol, Max_no_steps,     &
        Roundoff, Hmin, Hmax, Hstart,            &
        FacMin, FacMax, FacRej, FacSafe,         &
!  Error indicator
        IERR)

!~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
CONTAINS !  SUBROUTINES internal to Rosenbrock
!~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
   
!~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~   
 SUBROUTINE ros_ErrorMsg(Code,T,H,IERR)
!~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~   
!    Handles all error messages
!~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~   
  
   REAL(kind=dp), INTENT(IN) :: T, H
   INTEGER, INTENT(IN)  :: Code
   INTEGER, INTENT(OUT) :: IERR
   
   IERR = Code
   PRINT * , &
     'Forced exit from Rosenbrock due to the following error:' 
     
   SELECT CASE (Code)
    CASE (-1)    
      PRINT * , '--> Improper value for maximal no of steps'
    CASE (-2)    
      PRINT * , '--> Selected Rosenbrock method not implemented'
    CASE (-3)    
      PRINT * , '--> Hmin/Hmax/Hstart must be positive'
    CASE (-4)    
      PRINT * , '--> FacMin/FacMax/FacRej must be positive'
    CASE (-5) 
      PRINT * , '--> Improper tolerance values'
    CASE (-6) 
      PRINT * , '--> No of steps exceeds maximum bound'
    CASE (-7) 
      PRINT * , '--> Step size too small: T + 10*H = T', &
            ' or H < Roundoff'
    CASE (-8)    
      PRINT * , '--> Matrix is repeatedly singular'
    CASE DEFAULT
      PRINT *, 'Unknown Error code: ', Code
   END SELECT
   
   PRINT *, "T=", T, "and H=", H
     
 END SUBROUTINE ros_ErrorMsg

!~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 SUBROUTINE ros_Integrator (Y, Tstart, Tend, T,  &
        AbsTol, RelTol,                          &
!~~~> Integration parameters
        Autonomous, VectorTol, Max_no_steps,     &
        Roundoff, Hmin, Hmax, Hstart,            &
        FacMin, FacMax, FacRej, FacSafe,         &
!~~~> Error indicator
        IERR )
!~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
!   Template for the implementation of a generic Rosenbrock method
!      defined by ros_S (no of stages)
!      and its coefficients ros_{A,C,M,E,Alpha,Gamma}
!~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

  IMPLICIT NONE

!~~~> Input: the initial condition at Tstart; Output: the solution at T
   REAL(kind=dp), INTENT(INOUT) :: Y(N)
!~~~> Input: integration interval
   REAL(kind=dp), INTENT(IN) :: Tstart,Tend
!~~~> Output: time at which the solution is returned (T=Tend if success)
   REAL(kind=dp), INTENT(OUT) ::  T
!~~~> Input: tolerances
   REAL(kind=dp), INTENT(IN) ::  AbsTol(N), RelTol(N)
!~~~> Input: integration parameters
   LOGICAL, INTENT(IN) :: Autonomous, VectorTol
   REAL(kind=dp), INTENT(IN) :: Hstart, Hmin, Hmax
   INTEGER, INTENT(IN) :: Max_no_steps
   REAL(kind=dp), INTENT(IN) :: Roundoff, FacMin, FacMax, FacRej, FacSafe
!~~~> Output: Error indicator
   INTEGER, INTENT(OUT) :: IERR
! ~~~~ Local variables
   REAL(kind=dp) :: Ynew(N), Fcn0(N), Fcn(N)
   REAL(kind=dp) :: K(N*ros_S), dFdT(N)
#ifdef FULL_ALGEBRA    
   REAL(kind=dp) :: Jac0(N,N), Ghimj(N,N)
#else
   REAL(kind=dp) :: Jac0(LU_NONZERO), Ghimj(LU_NONZERO)
#endif
   REAL(kind=dp) :: H, Hnew, HC, HG, Fac, Tau
   REAL(kind=dp) :: Err, Yerr(N)
   INTEGER :: Pivot(N), Direction, ioffset, j, istage
   LOGICAL :: RejectLastH, RejectMoreH, Singular
!~~~>  Local parameters
   REAL(kind=dp), PARAMETER :: ZERO = 0.0_dp, ONE  = 1.0_dp
   REAL(kind=dp), PARAMETER :: DeltaMin = 1.0E-5_dp
!~~~>  Locally called functions
!    REAL(kind=dp) WLAMCH
!    EXTERNAL WLAMCH
!~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~


!~~~>  Initial preparations
   T = Tstart
   RSTATUS(Nhexit) = ZERO
   H = MIN( MAX(ABS(Hmin),ABS(Hstart)) , ABS(Hmax) )
   IF (ABS(H) <= 10.0_dp*Roundoff) H = DeltaMin

   IF (Tend  >=  Tstart) THEN
     Direction = +1
   ELSE
     Direction = -1
   END IF
   H = Direction*H

   RejectLastH=.FALSE.
   RejectMoreH=.FALSE.

!~~~> Time loop begins below

TimeLoop: DO WHILE ( (Direction > 0).AND.((T-Tend)+Roundoff <= ZERO) &
       .OR. (Direction < 0).AND.((Tend-T)+Roundoff <= ZERO) )

   IF ( ISTATUS(Nstp) > Max_no_steps ) THEN  ! Too many steps
      CALL ros_ErrorMsg(-6,T,H,IERR)
      RETURN
   END IF
   IF ( ((T+0.1_dp*H) == T).OR.(H <= Roundoff) ) THEN  ! Step size too small
      CALL ros_ErrorMsg(-7,T,H,IERR)
      RETURN
   END IF

!~~~>  Limit H if necessary to avoid going beyond Tend
   H = MIN(H,ABS(Tend-T))

!~~~>   Compute the function at current time
   CALL FunTemplate(T,Y,Fcn0)
   ISTATUS(Nfun) = ISTATUS(Nfun) + 1

!~~~>  Compute the function derivative with respect to T
   IF (.NOT.Autonomous) THEN
      CALL ros_FunTimeDerivative ( T, Roundoff, Y, &
                Fcn0, dFdT )
   END IF

!~~~>   Compute the Jacobian at current time
   CALL JacTemplate(T,Y,Jac0)
   ISTATUS(Njac) = ISTATUS(Njac) + 1

!~~~>  Repeat step calculation until current step accepted
UntilAccepted: DO

   CALL ros_PrepareMatrix(H,Direction,ros_Gamma(1), &
          Jac0,Ghimj,Pivot,Singular)
   IF (Singular) THEN ! More than 5 consecutive failed decompositions
       CALL ros_ErrorMsg(-8,T,H,IERR)
       RETURN
   END IF

!~~~>   Compute the stages
Stage: DO istage = 1, ros_S

      ! Current istage offset. Current istage vector is K(ioffset+1:ioffset+N)
       ioffset = N*(istage-1)

      ! For the 1st istage the function has been computed previously
       IF ( istage == 1 ) THEN
         !slim: CALL WCOPY(N,Fcn0,1,Fcn,1)
	 Fcn(1:N) = Fcn0(1:N)
      ! istage>1 and a new function evaluation is needed at the current istage
       ELSEIF ( ros_NewF(istage) ) THEN
         !slim: CALL WCOPY(N,Y,1,Ynew,1)
	 Ynew(1:N) = Y(1:N)
         DO j = 1, istage-1
           CALL WAXPY(N,ros_A((istage-1)*(istage-2)/2+j), &
            K(N*(j-1)+1),1,Ynew,1)
         END DO
         Tau = T + ros_Alpha(istage)*Direction*H
         CALL FunTemplate(Tau,Ynew,Fcn)
         ISTATUS(Nfun) = ISTATUS(Nfun) + 1
       END IF ! if istage == 1 elseif ros_NewF(istage)
       !slim: CALL WCOPY(N,Fcn,1,K(ioffset+1),1)
       K(ioffset+1:ioffset+N) = Fcn(1:N)
       DO j = 1, istage-1
         HC = ros_C((istage-1)*(istage-2)/2+j)/(Direction*H)
         CALL WAXPY(N,HC,K(N*(j-1)+1),1,K(ioffset+1),1)
       END DO
       IF ((.NOT. Autonomous).AND.(ros_Gamma(istage).NE.ZERO)) THEN
         HG = Direction*H*ros_Gamma(istage)
         CALL WAXPY(N,HG,dFdT,1,K(ioffset+1),1)
       END IF
       CALL ros_Solve(Ghimj, Pivot, K(ioffset+1))

   END DO Stage


!~~~>  Compute the new solution
   !slim: CALL WCOPY(N,Y,1,Ynew,1)
   Ynew(1:N) = Y(1:N)
   DO j=1,ros_S
         CALL WAXPY(N,ros_M(j),K(N*(j-1)+1),1,Ynew,1)
   END DO

!~~~>  Compute the error estimation
   !slim: CALL WSCAL(N,ZERO,Yerr,1)
   Yerr(1:N) = ZERO
   DO j=1,ros_S
        CALL WAXPY(N,ros_E(j),K(N*(j-1)+1),1,Yerr,1)
   END DO
   Err = ros_ErrorNorm ( Y, Ynew, Yerr, AbsTol, RelTol, VectorTol )

!~~~> New step size is bounded by FacMin <= Hnew/H <= FacMax
   Fac  = MIN(FacMax,MAX(FacMin,FacSafe/Err**(ONE/ros_ELO)))
   Hnew = H*Fac

!~~~>  Check the error magnitude and adjust step size
   ISTATUS(Nstp) = ISTATUS(Nstp) + 1
   IF ( (Err <= ONE).OR.(H <= Hmin) ) THEN  !~~~> Accept step
      ISTATUS(Nacc) = ISTATUS(Nacc) + 1
      !slim: CALL WCOPY(N,Ynew,1,Y,1)
      Y(1:N) = Ynew(1:N)
      T = T + Direction*H
      Hnew = MAX(Hmin,MIN(Hnew,Hmax))
      IF (RejectLastH) THEN  ! No step size increase after a rejected step
         Hnew = MIN(Hnew,H)
      END IF
      RSTATUS(Nhexit) = H
      RSTATUS(Nhnew)  = Hnew
      RSTATUS(Ntexit) = T
      RejectLastH = .FALSE.
      RejectMoreH = .FALSE.
      H = Hnew
      EXIT UntilAccepted ! EXIT THE LOOP: WHILE STEP NOT ACCEPTED
   ELSE           !~~~> Reject step
      IF (RejectMoreH) THEN
         Hnew = H*FacRej
      END IF
      RejectMoreH = RejectLastH
      RejectLastH = .TRUE.
      H = Hnew
      IF (ISTATUS(Nacc) >= 1)  ISTATUS(Nrej) = ISTATUS(Nrej) + 1
   END IF ! Err <= 1

   END DO UntilAccepted

   END DO TimeLoop

!~~~> Succesful exit
   IERR = 1  !~~~> The integration was successful

  END SUBROUTINE ros_Integrator


!~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
  REAL(kind=dp) FUNCTION ros_ErrorNorm ( Y, Ynew, Yerr, &
                               AbsTol, RelTol, VectorTol )
!~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
!~~~> Computes the "scaled norm" of the error vector Yerr
!~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
   IMPLICIT NONE

! Input arguments
   REAL(kind=dp), INTENT(IN) :: Y(N), Ynew(N), &
          Yerr(N), AbsTol(N), RelTol(N)
   LOGICAL, INTENT(IN) ::  VectorTol
! Local variables
   REAL(kind=dp) :: Err, Scale, Ymax
   INTEGER  :: i
   REAL(kind=dp), PARAMETER :: ZERO = 0.0_dp

   Err = ZERO
   DO i=1,N
     Ymax = MAX(ABS(Y(i)),ABS(Ynew(i)))
     IF (VectorTol) THEN
       Scale = AbsTol(i)+RelTol(i)*Ymax
     ELSE
       Scale = AbsTol(1)+RelTol(1)*Ymax
     END IF
     Err = Err+(Yerr(i)/Scale)**2
   END DO
   Err  = SQRT(Err/N)

   ros_ErrorNorm = MAX(Err,1.0d-10)

  END FUNCTION ros_ErrorNorm


!~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
  SUBROUTINE ros_FunTimeDerivative ( T, Roundoff, Y, &
                Fcn0, dFdT )
!~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
!~~~> The time partial derivative of the function by finite differences
!~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
   IMPLICIT NONE

!~~~> Input arguments
   REAL(kind=dp), INTENT(IN) :: T, Roundoff, Y(N), Fcn0(N)
!~~~> Output arguments
   REAL(kind=dp), INTENT(OUT) :: dFdT(N)
!~~~> Local variables
   REAL(kind=dp) :: Delta
   REAL(kind=dp), PARAMETER :: ONE = 1.0_dp, DeltaMin = 1.0E-6_dp

   Delta = SQRT(Roundoff)*MAX(DeltaMin,ABS(T))
   CALL FunTemplate(T+Delta,Y,dFdT)
   ISTATUS(Nfun) = ISTATUS(Nfun) + 1
   CALL WAXPY(N,(-ONE),Fcn0,1,dFdT,1)
   CALL WSCAL(N,(ONE/Delta),dFdT,1)

  END SUBROUTINE ros_FunTimeDerivative


!~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
  SUBROUTINE ros_PrepareMatrix ( H, Direction, gam, &
             Jac0, Ghimj, Pivot, Singular )
! --- --- --- --- --- --- --- --- --- --- --- --- ---
!  Prepares the LHS matrix for stage calculations
!  1.  Construct Ghimj = 1/(H*ham) - Jac0
!      "(Gamma H) Inverse Minus Jacobian"
!  2.  Repeat LU decomposition of Ghimj until successful.
!       -half the step size if LU decomposition fails and retry
!       -exit after 5 consecutive fails
! --- --- --- --- --- --- --- --- --- --- --- --- ---
   IMPLICIT NONE

!~~~> Input arguments
#ifdef FULL_ALGEBRA    
   REAL(kind=dp), INTENT(IN) ::  Jac0(N,N)
#else
   REAL(kind=dp), INTENT(IN) ::  Jac0(LU_NONZERO)
#endif   
   REAL(kind=dp), INTENT(IN) ::  gam
   INTEGER, INTENT(IN) ::  Direction
!~~~> Output arguments
#ifdef FULL_ALGEBRA    
   REAL(kind=dp), INTENT(OUT) :: Ghimj(N,N)
#else
   REAL(kind=dp), INTENT(OUT) :: Ghimj(LU_NONZERO)
#endif   
   LOGICAL, INTENT(OUT) ::  Singular
   INTEGER, INTENT(OUT) ::  Pivot(N)
!~~~> Inout arguments
   REAL(kind=dp), INTENT(INOUT) :: H   ! step size is decreased when LU fails
!~~~> Local variables
   INTEGER  :: i, ISING, Nconsecutive
   REAL(kind=dp) :: ghinv
   REAL(kind=dp), PARAMETER :: ONE  = 1.0_dp, HALF = 0.5_dp

   Nconsecutive = 0
   Singular = .TRUE.

   DO WHILE (Singular)

!~~~>    Construct Ghimj = 1/(H*gam) - Jac0
#ifdef FULL_ALGEBRA    
     !slim: CALL WCOPY(N*N,Jac0,1,Ghimj,1)
     !slim: CALL WSCAL(N*N,(-ONE),Ghimj,1)
     Ghimj = -Jac0
     ghinv = ONE/(Direction*H*gam)
     DO i=1,N
       Ghimj(i,i) = Ghimj(i,i)+ghinv
     END DO
#else
     !slim: CALL WCOPY(LU_NONZERO,Jac0,1,Ghimj,1)
     !slim: CALL WSCAL(LU_NONZERO,(-ONE),Ghimj,1)
     Ghimj(1:LU_NONZERO) = -Jac0(1:LU_NONZERO)
     ghinv = ONE/(Direction*H*gam)
     DO i=1,N
       Ghimj(LU_DIAG(i)) = Ghimj(LU_DIAG(i))+ghinv
     END DO
#endif   
!~~~>    Compute LU decomposition
     CALL ros_Decomp( Ghimj, Pivot, ISING )
     IF (ISING == 0) THEN
!~~~>    If successful done
        Singular = .FALSE.
     ELSE ! ISING .ne. 0
!~~~>    If unsuccessful half the step size; if 5 consecutive fails then return
        ISTATUS(Nsng) = ISTATUS(Nsng) + 1
        Nconsecutive = Nconsecutive+1
        Singular = .TRUE.
        PRINT*,'Warning: LU Decomposition returned ISING = ',ISING
        IF (Nconsecutive <= 5) THEN ! Less than 5 consecutive failed decompositions
           H = H*HALF
        ELSE  ! More than 5 consecutive failed decompositions
           RETURN
        END IF  ! Nconsecutive
      END IF    ! ISING

   END DO ! WHILE Singular

  END SUBROUTINE ros_PrepareMatrix


!~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
  SUBROUTINE ros_Decomp( A, Pivot, ISING )
!~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
!  Template for the LU decomposition
!~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
   IMPLICIT NONE
!~~~> Inout variables
#ifdef FULL_ALGEBRA    
   REAL(kind=dp), INTENT(INOUT) :: A(N,N)
#else   
   REAL(kind=dp), INTENT(INOUT) :: A(LU_NONZERO)
#endif
!~~~> Output variables
   INTEGER, INTENT(OUT) :: Pivot(N), ISING

#ifdef FULL_ALGEBRA    
   CALL  DGETRF( N, N, A, N, Pivot, ISING )
#else   
   CALL KppDecomp ( A, ISING )
   Pivot(1) = 1
#endif
   ISTATUS(Ndec) = ISTATUS(Ndec) + 1

  END SUBROUTINE ros_Decomp


!~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
  SUBROUTINE ros_Solve( A, Pivot, b )
!~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
!  Template for the forward/backward substitution (using pre-computed LU decomposition)
!~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
   IMPLICIT NONE
!~~~> Input variables
#ifdef FULL_ALGEBRA    
   REAL(kind=dp), INTENT(IN) :: A(N,N)
   INTEGER :: ISING
#else   
   REAL(kind=dp), INTENT(IN) :: A(LU_NONZERO)
#endif
   INTEGER, INTENT(IN) :: Pivot(N)
!~~~> InOut variables
   REAL(kind=dp), INTENT(INOUT) :: b(N)

#ifdef FULL_ALGEBRA    
   CALL  DGETRS( 'N', N , 1, A, N, Pivot, b, N, ISING )
   IF ( Info < 0 ) THEN
      PRINT*,"Error in DGETRS. ISING=",ISING
   END IF  
#else   
   CALL KppSolve( A, b )
#endif

   ISTATUS(Nsol) = ISTATUS(Nsol) + 1

  END SUBROUTINE ros_Solve



!~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
  SUBROUTINE Ros2
!~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
! --- AN L-STABLE METHOD, 2 stages, order 2
!~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

   IMPLICIT NONE
   DOUBLE PRECISION g

    g = 1.0_dp + 1.0_dp/SQRT(2.0_dp)
    rosMethod = RS2
!~~~> Name of the method
    ros_Name = 'ROS-2'
!~~~> Number of stages
    ros_S = 2

!~~~> The coefficient matrices A and C are strictly lower triangular.
!   The lower triangular (subdiagonal) elements are stored in row-wise order:
!   A(2,1) = ros_A(1), A(3,1)=ros_A(2), A(3,2)=ros_A(3), etc.
!   The general mapping formula is:
!       A(i,j) = ros_A( (i-1)*(i-2)/2 + j )
!       C(i,j) = ros_C( (i-1)*(i-2)/2 + j )

    ros_A(1) = (1.0_dp)/g
    ros_C(1) = (-2.0_dp)/g
!~~~> Does the stage i require a new function evaluation (ros_NewF(i)=TRUE)
!   or does it re-use the function evaluation from stage i-1 (ros_NewF(i)=FALSE)
    ros_NewF(1) = .TRUE.
    ros_NewF(2) = .TRUE.
!~~~> M_i = Coefficients for new step solution
    ros_M(1)= (3.0_dp)/(2.0_dp*g)
    ros_M(2)= (1.0_dp)/(2.0_dp*g)
! E_i = Coefficients for error estimator
    ros_E(1) = 1.0_dp/(2.0_dp*g)
    ros_E(2) = 1.0_dp/(2.0_dp*g)
!~~~> ros_ELO = estimator of local order - the minimum between the
!    main and the embedded scheme orders plus one
    ros_ELO = 2.0_dp
!~~~> Y_stage_i ~ Y( T + H*Alpha_i )
    ros_Alpha(1) = 0.0_dp
    ros_Alpha(2) = 1.0_dp
!~~~> Gamma_i = \sum_j  gamma_{i,j}
    ros_Gamma(1) = g
    ros_Gamma(2) =-g

 END SUBROUTINE Ros2


!~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
  SUBROUTINE Ros3
!~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
! --- AN L-STABLE METHOD, 3 stages, order 3, 2 function evaluations
!~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

   IMPLICIT NONE
   rosMethod = RS3
!~~~> Name of the method
   ros_Name = 'ROS-3'
!~~~> Number of stages
   ros_S = 3

!~~~> The coefficient matrices A and C are strictly lower triangular.
!   The lower triangular (subdiagonal) elements are stored in row-wise order:
!   A(2,1) = ros_A(1), A(3,1)=ros_A(2), A(3,2)=ros_A(3), etc.
!   The general mapping formula is:
!       A(i,j) = ros_A( (i-1)*(i-2)/2 + j )
!       C(i,j) = ros_C( (i-1)*(i-2)/2 + j )

   ros_A(1)= 1.0_dp
   ros_A(2)= 1.0_dp
   ros_A(3)= 0.0_dp

   ros_C(1) = -0.10156171083877702091975600115545E+01_dp
   ros_C(2) =  0.40759956452537699824805835358067E+01_dp
   ros_C(3) =  0.92076794298330791242156818474003E+01_dp
!~~~> Does the stage i require a new function evaluation (ros_NewF(i)=TRUE)
!   or does it re-use the function evaluation from stage i-1 (ros_NewF(i)=FALSE)
   ros_NewF(1) = .TRUE.
   ros_NewF(2) = .TRUE.
   ros_NewF(3) = .FALSE.
!~~~> M_i = Coefficients for new step solution
   ros_M(1) =  0.1E+01_dp
   ros_M(2) =  0.61697947043828245592553615689730E+01_dp
   ros_M(3) = -0.42772256543218573326238373806514_dp
! E_i = Coefficients for error estimator
   ros_E(1) =  0.5_dp
   ros_E(2) = -0.29079558716805469821718236208017E+01_dp
   ros_E(3) =  0.22354069897811569627360909276199_dp
!~~~> ros_ELO = estimator of local order - the minimum between the
!    main and the embedded scheme orders plus 1
   ros_ELO = 3.0_dp
!~~~> Y_stage_i ~ Y( T + H*Alpha_i )
   ros_Alpha(1)= 0.0_dp
   ros_Alpha(2)= 0.43586652150845899941601945119356_dp
   ros_Alpha(3)= 0.43586652150845899941601945119356_dp
!~~~> Gamma_i = \sum_j  gamma_{i,j}
   ros_Gamma(1)= 0.43586652150845899941601945119356_dp
   ros_Gamma(2)= 0.24291996454816804366592249683314_dp
   ros_Gamma(3)= 0.21851380027664058511513169485832E+01_dp

  END SUBROUTINE Ros3

!~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~


!~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
  SUBROUTINE Ros4
!~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
!     L-STABLE ROSENBROCK METHOD OF ORDER 4, WITH 4 STAGES
!     L-STABLE EMBEDDED ROSENBROCK METHOD OF ORDER 3
!
!      E. HAIRER AND G. WANNER, SOLVING ORDINARY DIFFERENTIAL
!      EQUATIONS II. STIFF AND DIFFERENTIAL-ALGEBRAIC PROBLEMS.
!      SPRINGER SERIES IN COMPUTATIONAL MATHEMATICS,
!      SPRINGER-VERLAG (1990)
!~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

   IMPLICIT NONE

   rosMethod = RS4
!~~~> Name of the method
   ros_Name = 'ROS-4'
!~~~> Number of stages
   ros_S = 4

!~~~> The coefficient matrices A and C are strictly lower triangular.
!   The lower triangular (subdiagonal) elements are stored in row-wise order:
!   A(2,1) = ros_A(1), A(3,1)=ros_A(2), A(3,2)=ros_A(3), etc.
!   The general mapping formula is:
!       A(i,j) = ros_A( (i-1)*(i-2)/2 + j )
!       C(i,j) = ros_C( (i-1)*(i-2)/2 + j )

   ros_A(1) = 0.2000000000000000E+01_dp
   ros_A(2) = 0.1867943637803922E+01_dp
   ros_A(3) = 0.2344449711399156_dp
   ros_A(4) = ros_A(2)
   ros_A(5) = ros_A(3)
   ros_A(6) = 0.0_dp

   ros_C(1) =-0.7137615036412310E+01_dp
   ros_C(2) = 0.2580708087951457E+01_dp
   ros_C(3) = 0.6515950076447975_dp
   ros_C(4) =-0.2137148994382534E+01_dp
   ros_C(5) =-0.3214669691237626_dp
   ros_C(6) =-0.6949742501781779_dp
!~~~> Does the stage i require a new function evaluation (ros_NewF(i)=TRUE)
!   or does it re-use the function evaluation from stage i-1 (ros_NewF(i)=FALSE)
   ros_NewF(1)  = .TRUE.
   ros_NewF(2)  = .TRUE.
   ros_NewF(3)  = .TRUE.
   ros_NewF(4)  = .FALSE.
!~~~> M_i = Coefficients for new step solution
   ros_M(1) = 0.2255570073418735E+01_dp
   ros_M(2) = 0.2870493262186792_dp
   ros_M(3) = 0.4353179431840180_dp
   ros_M(4) = 0.1093502252409163E+01_dp
!~~~> E_i  = Coefficients for error estimator
   ros_E(1) =-0.2815431932141155_dp
   ros_E(2) =-0.7276199124938920E-01_dp
   ros_E(3) =-0.1082196201495311_dp
   ros_E(4) =-0.1093502252409163E+01_dp
!~~~> ros_ELO  = estimator of local order - the minimum between the
!    main and the embedded scheme orders plus 1
   ros_ELO  = 4.0_dp
!~~~> Y_stage_i ~ Y( T + H*Alpha_i )
   ros_Alpha(1) = 0.0_dp
   ros_Alpha(2) = 0.1145640000000000E+01_dp
   ros_Alpha(3) = 0.6552168638155900_dp
   ros_Alpha(4) = ros_Alpha(3)
!~~~> Gamma_i = \sum_j  gamma_{i,j}
   ros_Gamma(1) = 0.5728200000000000_dp
   ros_Gamma(2) =-0.1769193891319233E+01_dp
   ros_Gamma(3) = 0.7592633437920482_dp
   ros_Gamma(4) =-0.1049021087100450_dp

  END SUBROUTINE Ros4

!~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
  SUBROUTINE Rodas3
!~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
! --- A STIFFLY-STABLE METHOD, 4 stages, order 3
!~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

   IMPLICIT NONE

   rosMethod = RD3
!~~~> Name of the method
   ros_Name = 'RODAS-3'
!~~~> Number of stages
   ros_S = 4

!~~~> The coefficient matrices A and C are strictly lower triangular.
!   The lower triangular (subdiagonal) elements are stored in row-wise order:
!   A(2,1) = ros_A(1), A(3,1)=ros_A(2), A(3,2)=ros_A(3), etc.
!   The general mapping formula is:
!       A(i,j) = ros_A( (i-1)*(i-2)/2 + j )
!       C(i,j) = ros_C( (i-1)*(i-2)/2 + j )

   ros_A(1) = 0.0_dp
   ros_A(2) = 2.0_dp
   ros_A(3) = 0.0_dp
   ros_A(4) = 2.0_dp
   ros_A(5) = 0.0_dp
   ros_A(6) = 1.0_dp

   ros_C(1) = 4.0_dp
   ros_C(2) = 1.0_dp
   ros_C(3) =-1.0_dp
   ros_C(4) = 1.0_dp
   ros_C(5) =-1.0_dp
   ros_C(6) =-(8.0_dp/3.0_dp)

!~~~> Does the stage i require a new function evaluation (ros_NewF(i)=TRUE)
!   or does it re-use the function evaluation from stage i-1 (ros_NewF(i)=FALSE)
   ros_NewF(1)  = .TRUE.
   ros_NewF(2)  = .FALSE.
   ros_NewF(3)  = .TRUE.
   ros_NewF(4)  = .TRUE.
!~~~> M_i = Coefficients for new step solution
   ros_M(1) = 2.0_dp
   ros_M(2) = 0.0_dp
   ros_M(3) = 1.0_dp
   ros_M(4) = 1.0_dp
!~~~> E_i  = Coefficients for error estimator
   ros_E(1) = 0.0_dp
   ros_E(2) = 0.0_dp
   ros_E(3) = 0.0_dp
   ros_E(4) = 1.0_dp
!~~~> ros_ELO  = estimator of local order - the minimum between the
!    main and the embedded scheme orders plus 1
   ros_ELO  = 3.0_dp
!~~~> Y_stage_i ~ Y( T + H*Alpha_i )
   ros_Alpha(1) = 0.0_dp
   ros_Alpha(2) = 0.0_dp
   ros_Alpha(3) = 1.0_dp
   ros_Alpha(4) = 1.0_dp
!~~~> Gamma_i = \sum_j  gamma_{i,j}
   ros_Gamma(1) = 0.5_dp
   ros_Gamma(2) = 1.5_dp
   ros_Gamma(3) = 0.0_dp
   ros_Gamma(4) = 0.0_dp

  END SUBROUTINE Rodas3

!~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
  SUBROUTINE Rodas4
!~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
!     STIFFLY-STABLE ROSENBROCK METHOD OF ORDER 4, WITH 6 STAGES
!
!      E. HAIRER AND G. WANNER, SOLVING ORDINARY DIFFERENTIAL
!      EQUATIONS II. STIFF AND DIFFERENTIAL-ALGEBRAIC PROBLEMS.
!      SPRINGER SERIES IN COMPUTATIONAL MATHEMATICS,
!      SPRINGER-VERLAG (1996)
!~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

   IMPLICIT NONE

    rosMethod = RD4
!~~~> Name of the method
    ros_Name = 'RODAS-4'
!~~~> Number of stages
    ros_S = 6

!~~~> Y_stage_i ~ Y( T + H*Alpha_i )
    ros_Alpha(1) = 0.000_dp
    ros_Alpha(2) = 0.386_dp
    ros_Alpha(3) = 0.210_dp
    ros_Alpha(4) = 0.630_dp
    ros_Alpha(5) = 1.000_dp
    ros_Alpha(6) = 1.000_dp

!~~~> Gamma_i = \sum_j  gamma_{i,j}
    ros_Gamma(1) = 0.2500000000000000_dp
    ros_Gamma(2) =-0.1043000000000000_dp
    ros_Gamma(3) = 0.1035000000000000_dp
    ros_Gamma(4) =-0.3620000000000023E-01_dp
    ros_Gamma(5) = 0.0_dp
    ros_Gamma(6) = 0.0_dp

!~~~> The coefficient matrices A and C are strictly lower triangular.
!   The lower triangular (subdiagonal) elements are stored in row-wise order:
!   A(2,1) = ros_A(1), A(3,1)=ros_A(2), A(3,2)=ros_A(3), etc.
!   The general mapping formula is:  A(i,j) = ros_A( (i-1)*(i-2)/2 + j )
!                  C(i,j) = ros_C( (i-1)*(i-2)/2 + j )

    ros_A(1) = 0.1544000000000000E+01_dp
    ros_A(2) = 0.9466785280815826_dp
    ros_A(3) = 0.2557011698983284_dp
    ros_A(4) = 0.3314825187068521E+01_dp
    ros_A(5) = 0.2896124015972201E+01_dp
    ros_A(6) = 0.9986419139977817_dp
    ros_A(7) = 0.1221224509226641E+01_dp
    ros_A(8) = 0.6019134481288629E+01_dp
    ros_A(9) = 0.1253708332932087E+02_dp
    ros_A(10) =-0.6878860361058950_dp
    ros_A(11) = ros_A(7)
    ros_A(12) = ros_A(8)
    ros_A(13) = ros_A(9)
    ros_A(14) = ros_A(10)
    ros_A(15) = 1.0_dp

    ros_C(1) =-0.5668800000000000E+01_dp
    ros_C(2) =-0.2430093356833875E+01_dp
    ros_C(3) =-0.2063599157091915_dp
    ros_C(4) =-0.1073529058151375_dp
    ros_C(5) =-0.9594562251023355E+01_dp
    ros_C(6) =-0.2047028614809616E+02_dp
    ros_C(7) = 0.7496443313967647E+01_dp
    ros_C(8) =-0.1024680431464352E+02_dp
    ros_C(9) =-0.3399990352819905E+02_dp
    ros_C(10) = 0.1170890893206160E+02_dp
    ros_C(11) = 0.8083246795921522E+01_dp
    ros_C(12) =-0.7981132988064893E+01_dp
    ros_C(13) =-0.3152159432874371E+02_dp
    ros_C(14) = 0.1631930543123136E+02_dp
    ros_C(15) =-0.6058818238834054E+01_dp

!~~~> M_i = Coefficients for new step solution
    ros_M(1) = ros_A(7)
    ros_M(2) = ros_A(8)
    ros_M(3) = ros_A(9)
    ros_M(4) = ros_A(10)
    ros_M(5) = 1.0_dp
    ros_M(6) = 1.0_dp

!~~~> E_i  = Coefficients for error estimator
    ros_E(1) = 0.0_dp
    ros_E(2) = 0.0_dp
    ros_E(3) = 0.0_dp
    ros_E(4) = 0.0_dp
    ros_E(5) = 0.0_dp
    ros_E(6) = 1.0_dp

!~~~> Does the stage i require a new function evaluation (ros_NewF(i)=TRUE)
!   or does it re-use the function evaluation from stage i-1 (ros_NewF(i)=FALSE)
    ros_NewF(1) = .TRUE.
    ros_NewF(2) = .TRUE.
    ros_NewF(3) = .TRUE.
    ros_NewF(4) = .TRUE.
    ros_NewF(5) = .TRUE.
    ros_NewF(6) = .TRUE.

!~~~> ros_ELO  = estimator of local order - the minimum between the
!        main and the embedded scheme orders plus 1
    ros_ELO = 4.0_dp

  END SUBROUTINE Rodas4
!~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
  SUBROUTINE Rang3
!~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
! STIFFLY-STABLE W METHOD OF ORDER 3, WITH 4 STAGES
!
! J. RANG and L. ANGERMANN
! NEW ROSENBROCK W-METHODS OF ORDER 3
! FOR PARTIAL DIFFERENTIAL ALGEBRAIC
!        EQUATIONS OF INDEX 1                                             
! BIT Numerical Mathematics (2005) 45: 761-787
!  DOI: 10.1007/s10543-005-0035-y
! Table 4.1-4.2
!~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

   IMPLICIT NONE

    rosMethod = RG3
!~~~> Name of the method
    ros_Name = 'RANG-3'
!~~~> Number of stages
    ros_S = 4

    ros_A(1) = 5.09052051067020d+00;
    ros_A(2) = 5.09052051067020d+00;
    ros_A(3) = 0.0d0;
    ros_A(4) = 4.97628111010787d+00;
    ros_A(5) = 2.77268164715849d-02;
    ros_A(6) = 2.29428036027904d-01;

    ros_C(1) = -1.16790812312283d+01;
    ros_C(2) = -1.64057326467367d+01;
    ros_C(3) = -2.77268164715850d-01;
    ros_C(4) = -8.38103960500476d+00;
    ros_C(5) = -8.48328409199343d-01;
    ros_C(6) =  2.87009860433106d-01;

    ros_M(1) =  5.22582761233094d+00;
    ros_M(2) = -5.56971148154165d-01;
    ros_M(3) =  3.57979469353645d-01;
    ros_M(4) =  1.72337398521064d+00;

    ros_E(1) = -5.16845212784040d+00;
    ros_E(2) = -1.26351942603842d+00;
    ros_E(3) = -1.11022302462516d-16;
    ros_E(4) =  2.22044604925031d-16;

    ros_Alpha(1) = 0.0d00;
    ros_Alpha(2) = 2.21878746765329d+00;
    ros_Alpha(3) = 2.21878746765329d+00;
    ros_Alpha(4) = 1.55392337535788d+00;

    ros_Gamma(1) =  4.35866521508459d-01;
    ros_Gamma(2) = -1.78292094614483d+00;
    ros_Gamma(3) = -2.46541900496934d+00;
    ros_Gamma(4) = -8.05529997906370d-01;


!~~~> Does the stage i require a new function evaluation (ros_NewF(i)=TRUE)
!   or does it re-use the function evaluation from stage i-1 (ros_NewF(i)=FALSE)
    ros_NewF(1) = .TRUE.
    ros_NewF(2) = .TRUE.
    ros_NewF(3) = .TRUE.
    ros_NewF(4) = .TRUE.

!~~~> ros_ELO  = estimator of local order - the minimum between the
!        main and the embedded scheme orders plus 1
    ros_ELO = 3.0_dp

  END SUBROUTINE Rang3
!~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

!~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
!   End of the set of internal Rosenbrock subroutines
!~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
END SUBROUTINE Rosenbrock
!~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~


!~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
SUBROUTINE FunTemplate( T, Y, Ydot )
!~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
!  Template for the ODE function call.
!  Updates the rate coefficients (and possibly the fixed species) at each call
!~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 USE aromatics_kpp_Parameters, ONLY: NVAR, LU_NONZERO
 USE aromatics_kpp_Global, ONLY: FIX, RCONST, TIME
 USE aromatics_kpp_Function, ONLY: Fun
 USE aromatics_kpp_Rates, ONLY: Update_SUN, Update_RCONST
!~~~> Input variables
   REAL(kind=dp) :: T, Y(NVAR)
!~~~> Output variables
   REAL(kind=dp) :: Ydot(NVAR)
!~~~> Local variables
   REAL(kind=dp) :: Told

   Told = TIME
   TIME = T
   CALL Update_SUN()
   CALL Update_RCONST()
   CALL Fun( Y, FIX, RCONST, Ydot )
   TIME = Told

END SUBROUTINE FunTemplate


!~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
SUBROUTINE JacTemplate( T, Y, Jcb )
!~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
!  Template for the ODE Jacobian call.
!  Updates the rate coefficients (and possibly the fixed species) at each call
!~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 USE aromatics_kpp_Parameters, ONLY: NVAR, LU_NONZERO
 USE aromatics_kpp_Global, ONLY: FIX, RCONST, TIME
 USE aromatics_kpp_Jacobian, ONLY: Jac_SP, LU_IROW, LU_ICOL
 USE aromatics_kpp_LinearAlgebra
 USE aromatics_kpp_Rates, ONLY: Update_SUN, Update_RCONST
!~~~> Input variables
    REAL(kind=dp) :: T, Y(NVAR)
!~~~> Output variables
#ifdef FULL_ALGEBRA    
    REAL(kind=dp) :: JV(LU_NONZERO), Jcb(NVAR,NVAR)
#else
    REAL(kind=dp) :: Jcb(LU_NONZERO)
#endif   
!~~~> Local variables
    REAL(kind=dp) :: Told
#ifdef FULL_ALGEBRA    
    INTEGER :: i, j
#endif   

    Told = TIME
    TIME = T
    CALL Update_SUN()
    CALL Update_RCONST()
#ifdef FULL_ALGEBRA    
    CALL Jac_SP(Y, FIX, RCONST, JV)
    DO j=1,NVAR
      DO i=1,NVAR
         Jcb(i,j) = 0.0_dp
      END DO
    END DO
    DO i=1,LU_NONZERO
       Jcb(LU_IROW(i),LU_ICOL(i)) = JV(i)
    END DO
#else
    CALL Jac_SP( Y, FIX, RCONST, Jcb )
#endif   
    TIME = Told

END SUBROUTINE JacTemplate

END MODULE aromatics_kpp_Integrator




! End of INTEGRATE function
! ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~


